The document discusses the Fourier transform, which relates a signal sampled in time or space to the same signal sampled in frequency. It explains the mathematical definition and provides an example of using a Fourier transform to convert time domain data to the frequency domain. Specifically, it uses a cosine wave as input data and calculates the Fourier transform to reveal a strong amplitude at the expected frequency component.

1. Fourier Transforms
IsaacLu

2. Conception
 The Fourier transformation is a mathematical formula that relates a signal
sampled in time or space to the same signal sampled in frequency.
 In signal processing, the Fourier transform can reveal important
characteristics of a signal, namely, its frequency components.
動圖

3. Background knowledge
 Euler’s formula
 𝑒 𝑖𝜃
= 𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃
 Euler’s formula tells us that if we take exponential to some number 𝜃 times i,
we will we get the point land on a circle with radius 1 rotating 𝜃 angle.
𝜃
1

4. Background knowledge
 𝑒2𝜋𝑖𝑡
means the vector rotating 360 degree per time(time = t).
 So when we want to change frequency of rotation, the formula can change to
𝑒2𝜋𝑖𝑓, for example 𝑓 = 1/10 means it make one full turn every 10 seconds.
 𝑒−2𝜋𝑖𝑓
if we throw a negative sign up into that exponent mean rotation
direction from counterclockwise to clockwise.
t=10(f = 1/10)
𝑒−2𝜋𝑖𝑓

5. Mathematical definition
 𝑓(𝑘 + 1) = 𝑗=0
𝑛−1
𝑔 𝑗 + 1 ∗ 𝑒−2𝜋𝑖∗𝑡[𝑗]∗𝑘
,
 𝑠. 𝑡. 0 ≤ j, k ≤ n − 1
Length of signal: n
Time vector: t
Sampling frequency: k
Number of time point: j
𝜔 = 𝑒−2𝜋𝑖𝑡

6. Example
𝑔 𝑗 = cos 2𝜋 ∗ 50 ∗ 𝑡[𝑗] + 1.2, this is our time domain data.
Length of signal j = 1000, t[0] = 0,t[1] = 0.001, …, t[999] = 0.999
Our propose is using Fourier transformation to transforms data
from time domain to frequency.

7. First step: create a 𝜔 base on Euler’s formula
𝜔 = 𝑒−2𝜋𝑖𝑡
Why we need to throw a negative sign
up into that exponent just because
the convention in the context of
Fourier transforms is to think about
rotating in the clockwise direction.

8. Example
 Second step: 𝑔 𝑗 ∗ 𝑒−2𝜋𝑖𝑡
 Reason : We want to make our wave(g(t)) winding and rotation follow complex
number 𝑒−2𝜋𝑖𝑡
and getting scaled up and down base on amplitude(振幅).
Figure of 𝑔 𝑡 ∗ 𝑤
point 1 to point 500, I didn’t
draw next 500 points because
it will make figure to
intensive.
𝜔 = 𝑒−2𝜋𝑖𝑡

9.  Third step: time 𝑒 𝑘 with our equation →
 𝑔 𝑗 + 1 ∗ 𝑒−2𝜋𝑖∗𝑡∗𝑘
, 0 ≤ k ≤ n − 1
 Reason: So far there is still nothing about frequency and how to change time
domain to frequency domain, so we time 𝑒 𝑘
with our equation, to import frequency
k into our equation
 How: we summarize N points of their coordinate then divide by N(FT does not
divide by N) t[j]=5
f 𝑘 + 1 =
𝑗=0
𝑛−1
𝑔 𝑗 + 1 ∗ 𝑒−2𝜋𝑖∗𝑡[𝑗]∗𝑘
, 0 ≤ k, j ≤ n − 1

10. So our equation can be separate into
But also we know0 ≤ k ≤ n − 1, so the equation can do more separate into
𝑔 1 ∗ 𝑒−2𝜋𝑖∗𝑡[0]∗0
+ 𝑔 1 ∗ 𝑒−2𝜋𝑖∗𝑡 0 ∗1
+ 𝑔 1 ∗ 𝑒−2𝜋𝑖∗𝑡[0]∗2
+ … + 𝑔 1 ∗ 𝑒−2𝜋𝑖∗𝑡[0]∗(𝑛−1)
𝑔 2 ∗ 𝑒−2𝜋𝑖∗𝑡 1 ∗0 + 𝑔 2 ∗ 𝑒−2𝜋𝑖∗𝑡 1 ∗1 + 𝑔 2 ∗ 𝑒−2𝜋𝑖∗𝑡[1]∗2 + … + 𝑔 2 ∗ 𝑒−2𝜋𝑖∗𝑡[1]∗(𝑛−1)
.
.
.
𝑔 𝑛 ∗ 𝑒−2𝜋𝑖∗𝑡 𝑛−1 ∗0
+ 𝑔 𝑛 ∗ 𝑒−2𝜋𝑖∗𝑡 𝑛−1 ∗1
+ 𝑔 𝑛 ∗ 𝑒−2𝜋𝑖∗𝑡[𝑛−1]∗2
+ … + 𝑔 𝑛 ∗ 𝑒−2𝜋𝑖∗𝑡[𝑛−1]∗(𝑛−1)
𝑗=0
𝑛−1
𝑔 𝑗 + 1 ∗ 𝑒−2𝜋𝑖∗𝑡[𝑗]∗𝑘
= 𝑔 1 ∗ 𝑒−2𝜋𝑖∗𝑡 0 ∗𝑘
+
𝑔 2 ∗ 𝑒−2𝜋𝑖∗𝑡 1 ∗𝑘
+
.
.
.
𝑔 𝑛 ∗ 𝑒−2𝜋𝑖∗𝑡 𝑛−1 ∗𝑘
Time
domain
Frequency
domain

11.  Third step: summarize points on the wound-up graph with different frequency
 Reason:
 How: we summarize N points of their coordinate then divide by N(FT does not
divide by N)
FT 𝑘 + 1 = 𝑡=0
𝑛−1
𝑔 𝑗 + 1 ∗ 𝑒−2𝜋𝑖∗𝑡∗𝑓
, 0 ≤ f ≤ n − 1

12.  Third step: summarize points on the wound-up graph with different frequency
 Reason: find x-coordinate for center of mass, and the x-coordinate center of mass
is the frequency intensity
 How: we summarize N points of their coordinate then divide by N(FT does not
divide by N)
𝑓 𝑘 + 1 =
𝑗=0
𝑛−1
𝑔 𝑗 + 1 ∗ 𝑒−2𝜋𝑖∗𝑡∗𝑘 , 0 ≤ k ≤ n − 1

13. Changing variable k will find out our figure changed base on
frequency.
𝑔 𝑗 + 1 ∗ 𝑒−2𝜋𝑖∗𝑡[𝑗]∗𝑘
k = 1, 5, 10, 20, 30, 40, 50
K=1 K=5 K=10 K=20 K=30 K=40 K=50
x-
coordinate
center of
mass
5.77E-17 6.39E-17 -9.81E-17 -9.81E-17 -9.37E-17 1.72E-16 0.5

14. Follow-up processing
 After those three steps we finish Fourier transformation, but we get a lot of complex number
in our example f(k) has 1000 complex number.
 How to use them?
1. We divide f(k) by N
 Reason: just finish what FT didn’t do.
2. We calculate absolute value of f(k)/N and time 2
 Reason: our complex number data of f(k) is symmetrical
and conjugated, so we can just use positive part and because
we ignore negative part so we need time 2.
 After finish this two steps the new data is like
figure on the right side. And because data is
symmetrical so we only look at data from
0~500 is fine.

15. Conclusion
From the beginning we only
use one cosine wave to be our
input data. From figure at
right hand side we only see
one strong amplitude.

16. reference
 https://www.youtube.com/watch?v=spUNpyF58BY&
 https://zh.wikipedia.org/zh-
tw/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2